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Deeper than primes

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doronshadmi said:
Since you do not understand present continuous in terms of "ever increasing", let us express it as "permanently changing", where the permanent (invariant) aspect is like pi among the collection of circles, and the change (variant) aspect is like the curvature among the collection of circles.
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Are all of your circles circles, Doron, or are some of your circles not circles?

You can try to dance around with the wording all you want but the tune still hasn’t changed.


doronshadmi said:
The present continuous state, as described above, is the fundamental property of any infinite collection of different objects.

For more details, please look at http://www.internationalskeptics.com/forums/showpost.php?p=6963549&postcount=14542 .
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Since you still don't understand or perhaps just don't want to accept that all natural numbers are natural numbers please explain to us specifically which natural numbers your think are "permanently changing" into, well, natural numbers. Doron in case your haven’t got it yet, that all natural numbers are natural numbers is your so called “(invariant) aspect” of all natural numbers and the only aspect relevant to them being members of the set of all natural numbers.

It seems now that you have abandoned your “increasing” for just “changing”. Could that be because you were unable (or just unwilling) to address this question…


The Man said:
By all means please explain to us the difference between increasing and decreasing with “no past (before) and no future (after)”?
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?

Not to worry I can rephrase to accommodate your change in lexicon.


So by all means please explain to us the difference between changing and unchanging with “no past (before) and no future (after)”?


Again you should try to learn English better. “permanently” is an adverb, a class of words that modify verbs. The verb being modified in your expression is “changing”. So "permanently changing" specifically refers to the act of changing being permanent. Not a combination of some permanent aspects and some changing aspects.
 
doronshadmi said:
You simply ignore this:
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You are the one simply ignoring that that isn't part of the proof.

You are the one simply ignoring that the "construction of D" assumes there is an existing bijection between N and P(N).


So, (1) you have been claiming it is part of Cantor's Theorem. It isn't. You fail. (2) You didn't pay attention to that fine detail, the definition for D requires an non-existence bijection. (3) You claim from repeated constructions of different D's from different non-existent bijections you are able to come up with one of these non-existent bijections.

So, you "prove" the impossible by assuming the very impossible thing you are trying to prove.


And after all that, you still cannot provide a bijection between the members of {A} and its power set.
 
jsfisher said:
However, you made a couple of claims, different from above, that are flatly untrue. You claimed that for any set S,

(1) a bijection exists between S and its power set, P(S).
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That amounts to a catastrophic collapse of reason and a supernova of ignorance in connection with Doron's insight into an argument in Wikipedia where the introduction clearly reads
In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can be seen to be true by a much simpler proof than that given below, since in addition to subsets of A with just one member, there are others as well, and since n < 2n for all natural numbers n.
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epix said:
That amounts to a catastrophic collapse of reason and a supernova of ignorance in connection with Doron's insight into an argument in Wikipedia where the introduction clearly reads
Click to expand...

You are just missing his remarkable insight. You see, Cantor's Theorem is proved by contradiction. Assume a construction is possible; derive a contradiction. Simple. But you see, if you simply repeat that impossible construction enough times, sooner or later you must get a different result, right?

And this fully explains the brilliance that is Doron.
 
jsfisher said:
You are just missing his remarkable insight. You see, Cantor's Theorem is proved by contradiction. Assume a construction is possible; derive a contradiction. Simple. But you see, if you simply repeat that impossible construction enough times, sooner or later you must get a different result, right?

And this fully explains the brilliance that is Doron.
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The most remarkable thing is that Doron would not use the nature of the arguments that kept Cantor teaching in Halle and not in Berlin where he wanted to go. Doron says that there is always a bijection between members of N and P(N) with the consequence |N| = |P(N)|. He sees
http://en.wikipedia.org/wiki/Cantor's_theorem
but is unable to follow on the clue provided within the article:

N...........P(N)

1 <---> {4,5}
2 <---> {1,2,3}
3 <---> {4,5,6}
4 <---> {1,3,5}
.
.
.

Set N is an infinite source of natural numbers and P(N) is an infinite source of subsets of N. The natural numbers in the N-bag are ordered 1, 2, 3, 4 . . . (you pick them that way) and the subsets {},{},... are randomly organized in the P(N)-bag. So the example above can be continued infinitely, coz you never run out of natural numbers or the subsets -- there always will be a bijection between one member of N and one member of P(N).

This is much better and understandable argument. After some 50 trillion years of bijecting, the story is the same: Ni <---> P(N)i. The only problem is that after those 50 trillion years, the demonstration still concerns the finite case, coz the latest bijection is known, even though Albert Einstein would show reservations due to his definition of insanity:
Insanity: doing the same thing over and over again and expecting different results.
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I'm not sure if Einstein and Cantor ever met.
 
Oops, I accidentally unsubscribed to this thread a few days ago. What did I miss? Has doronshadmi been right about anything yet?
 
The Man said:
Are all of your circles circles, Doron, or are some of your circles not circles?
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They are all circles, yet the exact amount of them is unsatisfied (it is permanently changing), or in other words, the set of all circles has no exact size, which is a fact that you ignore and as a result you do not understand the inherent property of "+1" (permanently next ...) of the concept of cardinality.

Here is another example of the unsatisfied exact amount of an infinite set.

{...} is the set of all differences.

{...} is a member of {...}, such that {{...},...} where {...} is different than the rest ,... of {{...},...}, but then

{{...},...} is a member of {{...},...} such that {{{...},...},{...},...} where {{...},...} is different than the rest ,{...},... of {{{...},...},{...},...} but then

... etc. ... ad infinitum ... such that the size of the set of all differences is inherently unsatisfied.
 
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jsfisher said:
You are just missing his remarkable insight. You see, Cantor's Theorem is proved by contradiction. Assume a construction is possible; derive a contradiction. Simple. But you see, if you simply repeat that impossible construction enough times, sooner or later you must get a different result, right?

And this fully explains the brilliance that is Doron.
Click to expand...
Eech contradiction provides an explicit P(N) member that is mapped with an explicit N member, such that

{} ↔ 1
{1,2,3,...} ↔ 2

and the rest of N members are mapped with the rest of the P(N) members between {} and {1,2,3,...}.

It is all done by using Cantors construction method where B
( B = { x ∈ A : x ∉ f(x) } ) ( please see http://en.wikipedia.org/wiki/Cantor's_theorem ) is a placeholder for explicit |P(N)| P(N) members that are mapped in a bijection with N members, which are members of N that is a proper subset of P(N).

As we all know, there is a bijection between a set and its proper subset among infinite sets, in the case of N ↔ P(N) the the proper subset is N members.

Another irrelevant conclusion, seen in http://en.wikipedia.org/wiki/Cantor's_theorem is:
Another way to think of the proof is that B, empty or non-empty, is always in the power set of A. For f to be onto, some element of A must map to B. But that leads to a contradiction: no element of B can map to B because that would contradict the criterion of membership in B, thus the element mapping to B must not be an element of B meaning that it satisfies the criterion for membership in B, another contradiction. So the assumption that an element of A maps to B must be false; and f can not be onto.
Click to expand...

It is irrelevant exactly as the contradiction of Cantor's "proof" does not prevent the construction of explicit P(N) members, and the bijection between explicit P(N) members and explicit N members.
 
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jsfisher said:
The fact the function, f(), does not / cannot possibly exist is no problem for you at all, is it Doron.
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EDIT:

The fact that B is a placeholder for explicit P(N) members (exactly because of the fact that f() does not / cannot possibly exist) does not say anything to you at all, is it jsfisher?
 
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epix said:
1. The set of natural numbers has infinite membership, but that is not an obstacle that would prevent counting and the disemination of the latest figure of that process.
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"latest" or "process" have nothing to do with infinite sets.

epix said:
2. The term "infinite bijection" is a classic Doronian expression, coz the prefix bi- means two, and "two" doesn't trail aroma of infinity -- that's for sure.
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Two infinite sets have "trail aroma of infinity -- that's for sure."
 
doronshadmi said:
EDIT:

The fact that B is a placeholder for explicit P(N) members (exactly because of the fact that f() does not / cannot possibly exist) does not say anything to you at all, is it jsfisher?
Click to expand...


Poor grammar aside, yes, it does say quite a lot. Since f() does not exist, if B is defined in terms of f(), then B does not exist either. At least that's the way it works outside of Doronetics. B is not a "placeholder" for anything. It doesn't exist. Moreover, invoking it for multiple "rounds" doesn't bring it into existence.

Doron, you have constructed a trivial bijection using an imaginary construction and produced a result irrelevant for the problem at hand. You keep wandering away from the point: You claimed a bijection exists between any set and its power set. You continue to fail in all attempts to demonstrate even one such example.
 
doronshadmi said:
"latest" or "process" have nothing to do with infinite sets.
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The idea that you are wrong begins to emerge in the mind of normally developing Homo sapiens around the age of 8. Here is why: The sequence of natural numbers defined as N doesn't have a bound and can progress infinitely according to n+1 axiom (Peano). Since every member of the progressive sequence of the naturals is distinct, the infinite sequence represents an infinite set. The natural numbers are also called "counting numbers"
http://mathworld.wolfram.com/CountingNumber.html
after the way they've been used since Homo erectus. Since counting is a process designed to answer a question regarding quantities, there surely is something called "the latest count."
http://www.swingstateproject.com/diary/7493/aksen-the-latest-count

The neural axons of Homo sapiens are suppossed to create path

/count/counting numbers/natural numbers/sequence of natural numbers/ N/set of natural numbers/infinite sets

but sometimes the neural association fails, as seen in you quote. Does it mean "back to the drawing board?"

But that would mean the devolution of human species back to time T-Gen0!

"What's the latest banana count?"

:jaw-dropp
I thought there were more of them affected.
No, Heavenly Father. It was just Doron.
 
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jsfisher said:
Poor grammar aside, yes, it does say quite a lot. Since f() does not exist, if B is defined in terms of f(), then B does not exist either.
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Wrong, B is defined by the opposite terms of f(), such that "B exists AND f() does not exits" is a true proposition.

You do not understand this part ( http://en.wikipedia.org/wiki/Cantor's_theorem ):
Thus there is no x such that f(x) = B; in other words, B is not in the image of f. Because B is in the power set of A, the power set of A has a greater cardinality than A itself.
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and the fact that B is a placeholder for |P(A)| P(A) explicit members, that are mapped with explicit A members, which are members of A that is a proper subset of P(A), where both A and P(A) are infinite sets, and it is well-known that there is a bijection between infinite sets and the members of some of their proper subset.

This fact is shown between, for example, the set of natural numbers and its proper subset of, for example, even numbers,

This fact is shown between, for example, the set of rational numbers and its proper subset of natural numbers (the n/1 form).

This fact is shown between, for example, the set P(N) and its proper subset of N members, such that, by using Cantor's construction method, one explicitly defines (without exceptional) the P(N) members, which enables him\her to define the bijection between the explicit P(N) members, and the explicit N members, where N is a proper subset of P(N), and such a bijection is a fact among infinite sets and any arbitrary given proper subset of them.
 
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epix said:
The idea that you are wrong begins to emerge in the mind of normally developing Homo sapiens around the age of 8. Here is why: The sequence of natural numbers defined as N doesn't have a bound and can progress infinitely according to n+1 axiom (Peano). Since every member of the progressive sequence of the naturals is distinct, the infinite sequence represents an infinite set. The natural numbers are also called "counting numbers"
http://mathworld.wolfram.com/CountingNumber.html
after the way they've been used since Homo erectus. Since counting is a process designed to answer a question regarding quantities, there surely is something called "the latest count."
http://www.swingstateproject.com/diary/7493/aksen-the-latest-count

The neural axons of Homo sapiens are suppossed to create path

/count/counting numbers/natural numbers/sequence of natural numbers/ N/set of natural numbers/infinite sets

but sometimes the neural association fails, as seen in you quote. Does it mean "back to the drawing board?"

But that would mean the devolution of human species back to time T-Gen0!

"What's the latest banana count?"

:jaw-dropp
I thought there were more of them affected.
No, Heavenly Father. It was just Doron.
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All you demonstrate is your serial step-by-step reasoning of the considered subject, by ignore the parallel observation of the considered subject, which is done at present continuous state, where "before" or "after" do not hold.

In other words, by using only serial step-by-step reasoning of the considered subject, one can't grasp propositions like "permanently changing" as a present continuous (parallel) observation.

The inability to use parallel observation in addition to serial observation, is the essence of the death of Evolution.
 
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